Overview of
Derivatives
funny clip on why you should learn calculus:
http://www.hark.com/clips/gzsvwhbsww-so-tomorrow-im-gonna-teach-you-calculus
Understanding Slope
Understanding Slope
The Definition of the Derivative
Computing Using the Definition
#1
Now try evaluating on your own!
Got your result? Now let's see how this
looks like graphically when using
GeoGebra.
Now, let’s do some graphical analysis of
the derivative.
#2a
Given the
graph of
f(x), graph
f’(x).
You have 30
seconds.
If you did not finish, don’t worry
because...
Here is the answer!
Oh yay! Your teacher is so cool. She gave
you the answer! But, you know what that
means…
On your paper, answer the following questions:
2b. Why do we need open circles at (-2,2)
(2, -2) and (3,1)?
2c. Why is there a straight line crossing
through y= -2 and not, say, y= 2?
2d. Why do we only have horizontal green
lines and not vertical green lines?
Let’s do one more graphical derivative analysis before
we move on.
Remember that to show that a function is not
differentiable at corners, we did so by showing that the
limit did not exist at these points.
At the time, we still had no knowledge of the rules of
derivatives. But now we do! Let's re-examine some of
these functions, but now we will use to rules of
derivatives to show its derivative does not exist.
Functions are not differentiable at cusps,
so let’s take the derivative and evaluate
at zero.
#3
On your paper, now YOU try the same
with f(x) = |x|.
You can do it!
Let’s now move on to the rules of
derivatives
Having worked through the Webercise
from our Derivatives Unit Plan website,
you have now completed several proofs.
Let's revisit one of these: the chain rule.
Recall that I said that the important part
of a proof is to explain why. Here is your
teacher's write up of this proof.
Now let’s fill in some gaps.
4a One of the hypothesis of the proof is that g(x) cannot
equal g(c). Why is it important to have this
condition? What would happen if g(x) = g(c)?
4b The proof states: “We will now multiply by a fraction
equivalent to the number one.” Is the fraction we
multiplied by really equal to one? Why are we allowed
to do this?
4c. The proof states: “By moving the denominators…”
Why are we allowed to do this?
Let’s compute!
Compute the 1st and 2nd derivative of the following
using only the power and constant rule:
Here is the algebra required to complete the
assigned task:
Let’s compute the same problems again.
This time, however, solve each problem
using one or more of the following: the
product rule, chain rule and/ or quotient
rule.
Of course, we will also use the power
rule.
Now that you’ve taken derivatives with only two of the
derivative rules (constant and power) and with all 5
rules,
10a. which way did you find easier? Doing algebra first,
then differentiating? Or differentiating, then
using algebra to simplify?
10b. List the pros and cons of doing each.
Finally, let’s see some applications to
physics problems!
Objects falling in a straight path
Falling balls!
http://www.youtube.com/watch?v=fd7D1LWzWmo
Alright, now let’s focus on one of the balls.
Remember that mathematicians live in a vacuum. Thus,
we will not account for the wind resistance.
A ball is dropped from a height of 100 feet. Its height s
(in feet) at time t (in seconds) is given by the position
function
s(t) = (-16t^2) +100
Answer the following questions on your paper:
11) In the position function, s(t) = (-16t^2) +100, where does the
“100” come from?
12) Why do we have a “-16” and not a “16”?
13) Find the average velocity at [1,2].
14) Find the instantaneous velocity at t =1.
15) What is the difference between “average velocity” and
“instantaneous velocity”? How are they computed differently?
16) Find the acceleration function. Write an explanation as to why
the result is constant. Then, explain it to your “left desk
neighbor.”
One final problem. You ready for it? I
know you are!
Objects falling in a parabolic path.
Launching a rocket
http://www.youtube.com/watch?v=7l8XFhWngaw
For the sake of the problem, we will say that the rocket
did NOT explode. Thus, it hit the ground at the end.
From the video, you can see the rocket had a parabolic
path. So, we will describe this path by
s(t) = 160t- 16t^2
Let’s do some analysis and computation.
17) In our previous problem, we had a negative value for s(t)
regardless of what we picked for t. Is this also true for our
current problem? Explain why in terms of physics, not math.
We will then discuss this as a class.
18) How high does the rocket go? (Think: what information do
we need in order to answer this?)
19) When does the rocket hit the ground? (Think: what
variable should we be solving for and using what function?)
20) What is the acceleration function?
So when or where is this used?
We can find it in the study of rockets, as we have seen in the last problem.
But it can also be used in the making of artillery! How high should you
position your weapon so that it will hit x feet away? Simple to calculate if
you have derivatives!
In chemistry, you can find the concentration of a particular element in a
product by computing and analyzing a few derivatives.
In economics, we use derivatives to compute for the marginal revenue and
marginal cost.
In short, while calculus originally had its most use in physics, it is now
applicable in all areas of studies!
Thank you for your focus and participation. I will ask you
to do one final thing: an open-ended survey.
On your paper, write down ANY lingering questions you
may have about the material we have looked at, even
if it’s something we didn’t talk about in the
presentation.
Also, please write down any suggestions on how I may
improve for the next presentation.
YOU GUYS AND DOLLS ARE AWESOME!